For likelihood problems, it is common to use Bayesian inference and deploy the MCMC toolkit. Unlike frequentist estimation, Bayesian analyses rely on sampling from a posterior distribution rather than optimization. Well known samplers include Gibbs and Metropolis-Hastings. Because they can be slow to converge, gradient-based samplers are increasingly popular with Metropolis-adjusted Langevin dynamics (MALA), Hamiltonian Monte-Carlo (HMC), or stochastic gradient Langevin dynamics (SGLD) algorithms. A comparison between rnr, rqn, and MALA is given in Section . Unlike Bayesian inference, this paper does not require the information matrix equality for valid inference. Convergence diagnostics used for MCMC algorithms such as trace plots or formal tests [1]}, [2]}, [3]} can also be used to monitor convergence of rnr  and rqn. Lastly, while the detailed balance condition guarantees the posterior is a stationary solution of a Metropolis-Hastings algorithm, convergence rate results typically assume a (nearly) concave log-posterior – similar to the convexity assumption in this paper.Recall that the detailed balance condition ensures that the random walk Metropolis-Hastings algorithm is ergodic but deriving the rate of convergence is more difficult. See e.g. [4]}, [5]}, [6]}. [7]} show how optimization can be used to sample from a Bayesian posterior with an intractable likelihood. Here, optimization is used for resampling.
